Download - Practica Calculo II
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1.- Se dan los puntos: A= (1,-2,-3), B= (2,-3,0), C= (3,1,-9) y D= (-1,1,-12).Calcular la d stanc a entre: A y C! B y D! A y D! C y D.
Sol:
d A , C =
( x2 x1 )2 +( y2 y1 )2 +( z2 z1 )2
d A , C = (3 1)2 +(1( 2 ))2 +(9 ( 3 ))2 = 4 +9 +36 = 49 = "
d B ,D = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d B ,D = ( 1 2 )2 +(1 ( 3 ))2 +(12 0 )2 = 9 +16 +144 = 169 = 13
d A , D = ( x2 x1 )2 +( y2 y1 )
2 +( z2 z1 )2
d A , D = ( 1 1 )2+(1( 2))2+( 12 ( 3 ))2 = 4 +9 +81 = 94 = 9.#9$3
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d C , D = ( x2 x1 )2 +( y2 y1 )
2 +( z2 z1 )2
d C , D = ( 1 3 )2 +(1 1)2 +(12 ( 9 ))2 = 16 +0 +9 = 25 = $
2.- De%ostrar &ue los puntos P 1
= (-2,',-3), P 2
= (',-3,-2), P 3
= (-3,-2,')son los rt ces de un tr *n+ulo e&u l*tero.
Sol:
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d P 1 , P 2 = ( x2 x1 )2 +( y2 y1 )
2 +( z2 z1 )2
d P 1 , P 2 = (4( 2))2+( 3 4 )2 +(2( 3 ))2 = 36 +49 +1 = 86 = 9.2"3#
d P 1 , P 3 = ( 3 ( 2 ))2 +( 2 4 )2 +(4 ( 3 ))2 = 1 +36 +49 = 86 = 9.2"3#
d P 2 , P 3 = ( 3 4 )2 +(2( 3))2 +(4 ( 2 ))2 = 49 +1 +36 = 86 = 9.2"3#
os puntos P1 = (-2, ',-3), P 2 = (',-3,-2), P3 = (-3,-2,') s son los
rt ces de un tr *n+ulo e&u l*tero.
or&ue sus d stanc as son +uales: d P 1 , P 2 = d P 1 , P 3 = d P 2 , P 3 = 86 = 9.2"3#
3.- De%ostrar &ue los puntos P 1 = (', 2, '), P 2 = (10, 2,-2), P 3 = (2, 0,
-') son los rt ces de un tr *n+ulo e&u l*tero.
Sol:
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d P 1 , P 2 = (10 4 )2 +(2 2 )2 +( 2 4 )2 = 36 +0 +36 = 72 = .' $2
d P 1 , P 3 = (2 4 )2 +(0 2 )2 +( 4 4 )2 = 4 +4 +64 = 72 = .' $2
d P 2 , P 3 = (2 10 )2 +(0 2 )2 +(4 ( 2 ))2 = 64 +4 +4 = 72 = .' $2
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os puntos P1 = (', 2, '), P 2 = (10, 2,-2), P 3 = (2, 0, -') s son los
rt ces de un tr *n+ulo e&u l*tero.
or&ue sus d stanc as son +uales: d P 1 , P 2 = d P 1 , P 3 = d P 2 , P 3 = 72 = .' $2
'.- De%ostrar &ue es /s sceles el tr *n+ulo cuyos rt ces son: A= (3,-1, 2)! B=(0,-',2) y C= (-3, 2, 1)
Sol:
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d A , B = (0 3 )2 +( 4 ( 1 ))2 +(2 2 )2 = 9 +9 +0 = 18 = '.2'2#
d A , C = ( 3 3 )2 +(2( 1 ))2 +(1 2 )2 = 36 +9 +1 = 46 = #." 23
d B ,C = ( 3 0 )2 +(2 ( 4 ))2 +(1 2 )2 = 9 +36 +1 = 46 = #." 23
os puntos A= (3,-1, 2)! B= (0,-',2) y C= (-3, 2, 1) s or%an un tr *n+ulo/s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.
ados /+uales: d A , C = dB ,C = 46 = #." 23
ado des +ual: d A, B = 18 = '.2'2#
$.- De%ostrar &ue los puntos P 1 = (3,-1,2), P 2 = (0,-',2), P 3 = (-3,2, 1)
son los rt ces de un tr *n+ulo /s sceles
Sol:
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d P 1 , P 2 = ( x2 x1 )2 +( y2 y1 )
2 +( z2 z1 )2
d P 1 , P 2 = (0 3 )2+( 4 ( 1 ))2 +(2 2 )2 = 9 +9 +0 = 18 = '.2'2#
d P 1 , P 3 = ( 3 3 )2 +(2 ( 1 ))2 +(1 2 )2 = 36 +9 +1 = 46 = #." 23
d P 2 , P 3 = ( 3 0)2 +(2( 4 ))2 +(1 2 )2 = 9 +36 +1 = 46 = #." 23
os puntos P1 = (3,-1,2), P 2 = (0,-',2), P 3 = (-3,2, 1) s or%an un
tr *n+ulo /s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.
ados /+uales: d P 1 , P 3 = d P 2 , P 3 = 46 = #." 23
ado des +ual: d P 1 , P 2 = 18 = '.2'2#
#.- De%ostrar &ue los puntos P 1 = (1,1, 2), P 2 = ($, , $), P 3 = (2,', 12)
son los rt ces de un tr *n+ulo /s sceles.
Sol:
d P 1 , P 2=
( x2 x1 )2 +( y2 y1 )2 +( z2 z1 )2
d P 1 , P 2 = (5 1)2 +(8 1 )2+(5 2)2 = 16 +49 +9 = 74 = .#023
d P 1 , P 3 = (2 1 )2 +(4 1 )2 +(12 2 )2 = 1 +9 +100 = 110 = 10.' 0
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d P 2 , P 3 = (2 5 )2 +(4 8 )2 +(12 5 )2 = 9 +16 +49 = 74 = .#023
os puntos P1 = (1,1, 2), P2 = ($, , $), P 3 = (2,', 12) s or%an un
tr *n+ulo /s sceles por&ue t enen 2 se+%entos +uales y 1 lado des +ual.
ados /+uales: d P 1 , P 2 = d P2 , P 3 = 74 = .#023
ado des +ual: d P 1 , P 3 = 110 = 10.' 0
".- De%ostrar &ue los puntos P 1 = (1,-1, 3), P2 = (2,1, "), P 3 = (',2, #)
son los rt ces de un tr *n+ulo rect*n+ulo.
Sol:
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d P 1 , P 2 = (2 1 )2 +(1 ( 1 ))2 +(7 3 )2 = 1 +4 +16 = 21 = '.$ 2$
d P 1 , P 3 = (4 1 )2 +(2 ( 1 ))2 +(6 3 )2 = 9 +9 +9 = 27 = $.19#1
d P 2 , P 3 = (4 2 )2 +(2 1 )2 +(6 7 )2 = 4 +1+1 = 6 = 2.''9'
ara de%ostrar &ue es un tr *n+ulo rect*n+ulo usare%os el eore%a det*+oras:
Hipotenusa 2 = cateto 12 cateto 2 2 -4 ( 27 )
2
= ( 21 )2 +( 6 )2 -4 2"=2"
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.- De%ostrar &ue los puntos P 1 = (3,-1,#), P 2 = (-1, ",-2), P 3 = (1,-3,2)
son los rt ces de un tr *n+ulo rect*n+ulo.
Sol:
d P 1 , P 2 = ( x2 x1)
2
+( y2 y1)2
+( z2 z1 )2
d P 1 , P 2 = ( 1 3 )2 +(7 ( 1 ))2 +(2 6 )2 = 16 +64 +64 = 144 = 12
d P 1 , P 3 = (1 3 )2 +(3 ( 1 ))2 +(2 6 )2 = 4 +4 +16 = 24 = '. 9 9
d P 2 , P 3 = (1 ( 1 ))2 +( 3 7 )2 +(2 ( 2 ))2 = 4 +100 +16 = 120 = 10.9$''
ara de%ostrar &ue es un tr *n+ulo rect*n+ulo usare%os el eore%a det*+oras:
Hipotenusa 2 = cateto 12 cateto 2 2 -4 ( 144 )
2
= ( 24 )2+( 120 )2 -4
1''=1''
9.- 5allar el alor de A s la d stanc a entre los puntos P 1 y P 2 es 3 6 .
P1 = (-1,2, 3), P 2 = (-1, A, 2).
Sol:
d P 1 , P 2 = 3 6 = ( 1 ( 1 ))2 +( A 2 )2 +(2 3 )2
3 6 = 0 +( A 2 )2 +1 -4 (
3 62 = (
( A 2 )2 +12
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96(#)= A2
- 'A ' 1 -4 $'-$= A2
- 'A -4 A2
- 'A 7 '9=0
A1,2 =( 4 ) ( 4 )2 4 (1)( 49 )
2 (1 ) =4 16 +196
2 =4 212
2
8ntonces: A1 =4 + 212
2 -4 A1 =2 53 -4 A1 =9.2 01
A2 =4 212
2 -4 A2 =2- 53 -4 A2 =-$.2 01
8l punto &uedar a P2 = (-1, 2 53 , 2)
10.- 5allar el per %etro del tr *n+ulo cuyos rt ces son A=(-2,-3,-2), B=(-3,1,'), C=(2,3,-1)
Sol:
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d A , B = ( 3 ( 2 ))2 +(1 ( 3 ))2 +(4 ( 2 ))2 = 1+16 +36 = 53 = ".2 01
d A , C =
(2( 2))2+(3 ( 3 ))2 +(1 ( 2 ))2=
16 +36 +1=
53 = ".2 01
d B ,C = (2 ( 3 ))2 +(3 1 )2 +(1 4 )2 = 25 +4 +25 = 54 = ".3' '
ene%os un tr *n+ulo /s sceles y su per %etro ser a:
er %etro= 53 53 54 -4 er %etro = 21.90 #
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11.- 5allar la d stanc a del punto (3,-#,$) a cada uno de los planos y e escoordenados
Sol:
;= (3,0, 0),
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12.- 5allar la d stanc a del punto ($, ',-3) a cada uno de los planos y e escoordenados.
Sol:
;= ($,0, 0),
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>= (%?, %y, %@)
%?=5 +1
2 -4 %?=3, %y= 2 +(7 )
2 -4 %y=-'.$, %@= 3 +4
2 -4
%@=0.$
8l punto %ed o entre P 1 = ($,-2,-3) y P 2 = (1,-",') es: >= (3,-'.$, 0.$)
1'.- no de los e?tre%os de un se+%ento de lon+ tud $ es el punto (-3, 2, -1)s las coordenadas ?, y, del otro e?tre%o son $,3 respect a%ente allar lacoordenada @.
Sol:
P1 = (-3, 2,-1), P 2 = ($,3, @)
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d P 1 , P 2 = $ = (5 ( 3 ))2 +(3 2 )2 +( z( 1 ))2
$ = 64 +1 +( z+1 )2
-4 (5
2 = ( 65 +( z+1 )2
2
2$=#$ z2
2@ 1 -4 2$-##= z2
2@ -4 z2
2@ '1=0
z1,2 = ( 2) (2)2 4 (1)(41 )
2 (1) = 2 4 164
2 = 2 160
2 = 2 4 10 i
2
8ntonces: z1 = 2 +4 10 i
2
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z2 = 2 4 10 i
2
o e? ste soluc n para n %eros reales
1$.- 5allar en el e e de aEsc sas un punto cuya d stanc a al punto A= (-3,', )sea +ual a 12.
Sol:
8n el e e de aEsc sas (e e ?) =(?, 0, 0), d P , A =12
d P 1 , P 2 = ( x2 x1)2 +( y2 y1)
2 +( z2 z1 )2
d P , A =12 = ( 3 x)2
+(4 0 )2
+(8 0 )2
-4 (12
2 = ( ( 3 x)2 +16 +64
2
1''= 9+6 x+ x2+80 -4 x
2+6 x 55 = 0
x1 , 2=( 6 ) (6 )2 4 (1)( 55 )
2 (1 ) = 6 36 +220
2 = 6 256
2 = 6 16
2 =
3 8
x1 = 3 +8 -4 x1 = 5
x2 = 3 8 -4 x2= 11
S ree%pla@a%os x1 = 5 y x2= 11 en: 12= ( 3 x)2+16 +64
12= ( 3 5 )2 +16 +64 cu%ple &ue 12=12
12= ( 3 ( 11 ))2 +16 +64 cu%ple &ue 12=12
8? sten dos puntos en el e e de las aEsc sas &ue cu%plen la cond c n ded stanc a de 12
P1 = ($,0, 0) y P2=( 11,0,0 )
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1#.- 5allar en el e e de ordenadas un punto e&u d stante a los puntos A= (1,-3,"), B= ($,", -$)
Sol:
8n el e e de ordenadas (e e y) = (0, y, 0),d P , A
=d P ,B
=F
d P , A = (1 0 )2 +(3 y)2 +(7 0 )2 = 1 +( 3 y)
2 +49 = ( 3 y)2 +50
d P , A = ( 3 y)2 +50
d P ,B = (5 0 )2 +(7 y)2 +(5 0 )2 = 25 +(7 y)
2 +25 = (7 y)2 +50
d P ,B=
(7 y)2 +50
/+ualando por la cond c n del proEle%a: d P , A = d P ,B
( 3 y)2 +50 = (7 y)2 +50
( ( 3 y)2 +50
2 = ( (7 y)2 +50
2
( 3 y)2+50 = (7 y)2 +50 -4 9 #y y
2
='9-1'y y2
-4 20y='0 -4
y=2
5allando la d stanc a d P , A y d P ,B
d P , A = 1+( 3 y)2 +49 = 1+( 3 2 )
2 +49 -4 d P , A= 75
d P ,B = 25 +(7 y)2 +25 = 25 +(7 2 )
2 +25 -4 d P ,B = 75
G nal%ente d P , A = d P ,B ="$ y el punto en las e es de las ordenadas es: =
(0,2, 0)
1".- Dados los rt ces de un tr *n+ulo A= (3, 2,-$), B= (1,-', 3), C= (-3,0, 1).5allar los puntos %ed os de sus lados.
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Sol:
>= ( xm, ym, zm ) a ser punto %ed o entre AB
xm =
3 +12 =
2 ,
ym =
2 +( 4 )2 = 1 ,
zm =
5 +32 =
1 -4 >= (2,-1,
-1)
= ( xn , yn , zn ) a ser punto %ed o entre AC
xn =3 +( 3 )
2= 0 , yn =
2 +02
= 1 , zn = 5 +1
2= 2 -4 = (0, 1, -2)
= ( x p , y p , z p ) a ser punto %ed o entre BC
x p =1 +( 3)
2= 1 , y p =
4 +02
= 2 , z p =3 +1
2= 2 -4 = (-1, -2,
2)